We have explicitly described the structure of rational WZ-pairs in terms of special pairs.
With structure theorems, we can easily generate rational WZ-pairs, which solves Problem 
in the rational case completely.
For the future research, the next direction is to solve
the problem in the cases of more general functions.
Using the terminology of Gessel in [1]}, a hypergeometric term \(F(x, y)\)  is said to be a WZ-function if
there exists another hypergeometric term \(G(x, y)\)  such that \((F, G)\)  is a WZ-pair. In the scheme of creative telescoping,
\((F, G)\)  being a WZ-pair with respect to \((\partial _x, \partial _y)\)  is equivalent to that
\(\partial _x\)  being a telescoper for \(F\)  with certificate \(G\) . Complete criteria for the
existence of telescopers for hypergeometric terms and their variants are known [2]}, [3]}, [4]}.
With the help of existence criteria for telescopers, one can show that \(F(x, y)\)  can be decomposed as the sum
\(F = \partial _y(H_1) + H_2\)  with \(H_1, H_2\)  being hypergeometric terms and \(H_2\)  is of proper form (see definition in [5]}, [1]})
if \(F\)  is a WZ-function.
So it is promising to apply the ideas in the study of the existence problem of telescopers to explore the structure of WZ-pairs.
